Trapped-ion entangling gates with bichromatic pair of microwave fields and magnetic field gradient

ABSTRACT

A trapped-ion quantum logic gate and a method of operating the trapped-ion quantum logic gate are provided. The trapped-ion quantum logic gate includes at least one ion having two internal states and forming a qubit having a qubit transition frequency ω0, a magnetic field gradient, and two microwave fields. Each of the two microwave fields has a respective frequency that is detuned from the qubit transition frequency ω0 by frequency difference δ. The at least one ion has a Rabi frequency Ωμ due to the two microwave fields and a Rabi frequency Ωg due to the magnetic field gradient. The method includes applying the magnetic field gradient and the two microwave fields to the at least one ion such that a quantity Ωg/δ is in a range between zero and 5×10−2.

CLAIM OF PRIORITY

This application claims the benefit of priority to U.S. Provisional Appl. No. 62/924,628, filed on Oct. 22, 2019 and incorporated in its entirety by reference herein.

STATEMENT REGARDING FEDERALLY SPONSORED R&D

The United States Government has rights in this application pursuant to Contract No. DE-AC52-07NA27344 between the United States Department of Energy and Lawrence Livermore National Security, LLC for the operation of Lawrence Livermore National Laboratory.

BACKGROUND Field

This application relates to quantum information devices utilizing trapped ions as qubits.

Description of the Related Art

Due to their inherent uniformity and exceptional coherence properties, trapped ions are a promising platform for scalable quantum simulations and general purpose quantum computing (see, e.g., Cirac 1995: Phys. Rev. Lett. 74 4091; Monroe 1995: Phys. Rev. Lett. 75 4714; Nielsen 2010: Quantum computation and quantum information (Cambridge University Press); Haffner 2008: Phys. Rep. 469 155; Blatt 2008: Nature 453 1008; Ladd 2010: Nature 464 45; Harty 2014: Phys. Rev. Lett. 113, 220501). Quantum entanglement, a necessary component of these two applications, is created in the ions' internal degrees of freedom (e.g., qubit “spin” states) via coupling to shared motional degrees of freedom (e.g., motional modes) (see, e.g., Wineland 1998: J. Res. Natl. Inst. Stand. and Technol. 103, 259). This spin-motion coupling can be achieved with one or more spatially dependent electromagnetic fields. One challenge for trapped-ion quantum logic is obtaining robust, scalable methods for spin-motion coupling with minimal error.

Geometric phase gates, which create entanglement through closed spin-dependent trajectories in motional phase space, are widely used because (in the Lamb-Dicke limit) they are first-order insensitive to ion temperature (see, e.g., Mølmer 1999: Phys. Rev. Lett. 82 1835; Sørensen 2000: Phys. Rev. A 62 022311; Leibfried 2003: Nature 422 412). Geometric phase gates employing laser beams and hyperfine qubits to create the spin-motion coupling have been used to generate Bell states with fidelities ˜0.999 (see, e.g., Ballance 2016: Phys. Rev. Lett. 117 060504; Gaebler 2016: Phys. Rev. Lett. 117 060505). In such laser-based schemes, two interfering non-copropagating laser beams create a moving optical lattice, whose state-dependent force couples the ions' internal degrees of freedom to their shared motion. The reported dominant errors were due to off-resonant photon scattering (see, e.g., Ozeri 2007: Phys. Rev. A 75 042329).

SUMMARY

In certain implementations, a method of operating a trapped-ion quantum logic gate is provided. The method comprises providing a trapped-ion quantum logic gate, the gate comprising at least one ion having two internal states and forming a qubit having a qubit transition frequency ω₀, a magnetic field gradient, and two microwave fields. Each of the two microwave fields has a respective frequency that is detuned from the qubit transition frequency ω₀ by frequency difference δ. The at least one ion has a Rabi frequency Ω_(μ) due to the two microwave fields and a Rabi frequency Ω_(g) due to the magnetic field gradient. The method further comprises applying the magnetic field gradient and the two microwave fields to the at least one ion such that a quantity Ω_(g)/δ is in a range between zero and 5×10⁻².

In certain implementations, a trapped-ion quantum logic gate is provided. The gate comprises at least one ion having two internal states and forming a qubit having a qubit transition frequency ω₀. The gate further comprises a magnetic field gradient, and two microwave fields. Each of the two microwave fields has a respective frequency that is detuned from the qubit transition frequency ω₀ by frequency difference δ. The at least one ion has a Rabi frequency Ω_(μ) due to the two microwave fields and a Rabi frequency Ω_(g) due to the magnetic field gradient. The magnetic field gradient and the two microwave fields are configured such that a quantity Ω_(g)/δ is in a range between zero and 5×10⁻².

In certain implementations, an intrinsic dynamically decoupled microwave-based zz (“IDD ZZ”) entangling gate is provided. The IDD ZZ gate is configured to be insensitive to qubit frequency shifts automatically. The IDD ZZ gate comprises trapped ions, a magnetic field gradient, and two microwave fields. The IDD ZZ gate does not require added fields to dynamically decouple itself from noise sources or to align the trapped ions. The IDD ZZ gate is configured to commute with static errors caused by miscalibrations and to use the two microwave fields to perform dynamical decoupling.

In certain implementations, a method of intrinsic dynamically decoupled microwave-based zz (“IDD ZZ”) trapped-ion entangling gating is provided. The method is insensitive to qubit frequency shifts automatically. The IDD ZZ gate comprises trapped ions, a magnetic field gradient, and two microwave fields. The IDD ZZ gate does not require added fields to dynamically decouple itself from noise sources or to align the trapped ions. The IDD ZZ gate is configured to commute with static errors caused by miscalibrations and to use the two microwave fields to perform dynamical decoupling.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a plot of the relative strengths of the gate Rabi frequencies corresponding to the first three resonances versus 4Ω_(μ)/δ in accordance with certain implementations described herein.

FIGS. 2A and 2B are plots of the gate fidelity

, in both the bichromatic interaction picture and the ion frame, for the YY gate operation with no pulse shaping and with pulse shaping, respectively, in accordance with certain implementations described herein.

FIG. 3A is a plot of the gate fidelity of a YY gate and a ZZ gate in accordance with certain implementations described herein.

FIG. 3B is a plot of the gate infidelity 1−

of a YY gate in accordance with certain implementations described herein.

FIG. 4 is a plot of the gate infidelity 1−

of a ZZ gate in accordance with certain implementations described herein.

FIG. 5A is a plot of the frequency spectrum of two microwave fields (frequencies: ω₀+δ and ω₀−δ), a magnetic field gradient field (frequency: ω_(g)), a frequency ω_(r) of a shared collective motional mode, and a qubit transition frequency ω₀, in accordance with certain implementations described herein.

FIG. 5B is a plot of the motional phase space trajectories of two gates that are insensitive to qubit frequency shifts for a conventional single-tone gate (dashed line) and a motionally robust gate (solid line) in accordance with certain implementations described herein.

FIG. 5C is a plot of the amplitudes of Bessel functions J_(n) as a function of frequency in the rotating frame of the qubit in accordance with certain implementations described herein.

FIG. 6A is a plot of numerical simulations of the impact of static shifts ε on the gate infidelity 1−

in accordance with certain implementations described herein.

FIG. 6B is a plot of numerical simulations of the sensitivity of the gate infidelity 1−

to oscillating qubit frequency shifts in accordance with certain implementations described herein.

FIG. 7A is a logarithmic plot of the gate infidelity 1−

as a function of the motional frequency offset ν for both of the qubit-frequency shift-resistant gates of FIGS. 6A and 6B, with a fixed gradient Ω_(g), in accordance with certain implementations described herein.

FIG. 7B is a plot of the Bell state infidelity for various intrinsically dynamically decoupled gates in accordance with certain implementations described herein.

FIG. 8A is a plot of the gate infidelity due to motional heating versus normalized gate speed 2πΩ_(g)t_(G) in accordance with certain implementations described herein.

FIG. 8B is a plot of the same calculations as FIG. 8A, but for motional dephasing instead of motional heating, assuming a motional dephasing rate of Γ_(d)=Ω_(g)/100π, in accordance with certain implementations described herein.

FIG. 9 schematically illustrates an example surface electrode configuration for an example trapped-ion quantum logic gate compatible with certain implementations described herein.

DETAILED DESCRIPTION

Overview

Alternative laser-free schemes use microwaves and induce spin-motion coupling with static magnetic field gradients (see, e.g., Mintert 2001: Phys. Rev. Lett. 87 257904; Khromova 2012: Phys. Rev. Lett. 108 220502; Lake 2015: Phys. Rev. A 91 012319; Randall 2015: Phys. Rev. A 91, 012322; Weidt 2016: Phys. Rev. Lett. 117 220501; Webb 2018: Phys. Rev. Lett. 121, 180501 (2018)), near-qubit-frequency magnetic field gradients (see, e.g., Ospelkaus 2008: Phys. Rev. Lett. 101 090502; Ospelkaus 2011: Nature 476 181; Harty 2016: Phys. Rev. Lett. 117 140501; Wolk 2017: New J. Phys. 19 083021; Hahn 2019: New J. Phys. Quantum Information 5, 70; Zarantonello 2019: arXiv preprint arXiv:1911.03954), or near-motional-frequency magnetic field gradients (see, e.g., Ospelkaus 2008; Chiaverini 2008: Phys. Rev. A 77 022324; Sutherland 2019: New J. Phys. 21, 033033; Srinivas 2019: Phys. Rev. Lett. 122 163201).

Laser-free schemes are not limited by photon scattering errors, do not require stable, high power lasers, and phase control is significantly easier than in the optical domain. Furthermore, microwave and rf sources are readily scalable to meet the requirements of larger quantum processors.

Such laser-free schemes do have some disadvantages. For example, schemes which employ static magnetic field gradients to perform laser-free gates can provide high-fidelity individual addressing of the ions in frequency space (see, e.g, Johanning 2009: Phys. Rev. Lett. 102, 073004), the microwave and rf frequencies used to realize an entangling gate are different for each ion, increasing the total number of drive tones for implementing a gate (see, e.g., Weidt 2016; Webb 2018). In addition, laser-free schemes can be slower than laser-based grates (e.g., by an order of magnitude) so the qubits spend more time entangled with the motional mode, so the gates can be more susceptible to other noise sources.

Qubit frequency shifts or miscalibrations due to fluctuating field amplitudes are significant sources of error in laser-free gates implemented with microwave field gradients (see, e.g., Webb 2018; Ospelkaus 2011). Some of these shifts may be reduced passively through careful trap design (see, Hahn 2019). Recently, a microwave-based Mølmer-Sørenson ({circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ), where {circumflex over (σ)}_(ϕ)≡{circumflex over (σ)}_(x) cos ϕ+{circumflex over (σ)}_(y) sin ϕ) entangling gate (see, e.g., Mølmer 1999; Sørensen 2000; Roos 2008: New J. Phys. 10 013002) was demonstrated with a fidelity of approximately 0.997 (see, Harty 2016). This high-fidelity microwave gate, which utilizes magnetic field gradients oscillating close to the qubit frequency, was achieved using an additional dynamical decoupling field (see, e.g., Viola 1998: Phys. Rev. A 58 2733; Viola 1999: Phys. Rev. Lett. 82 2417; Bermudez 2012: Phys. Rev. A 85 040302; Tan 2013: Phys. Rev. Lett. 110 263002) to suppress errors due to qubit frequency fluctuations, a significant source of decoherence in the system. However, the dynamical decoupling demonstrated in Harty 2016 utilized an extra field that is separate from, and commutes with, the gate Hamiltonian, with the precision phase control of an additional field increasing the experimental complexity. While decoherence from secular frequency shifts can be suppressed with Walsh sequences (see, Hayes 2012: Phys. Rev. Lett. 109, 020503) or with phase modulation of gate fields [Green 2015: Phys. Rev. Lett. 114, 120502; Leung 2018: Phys. Rev. Lett. 120, 020501), these techniques increase the gate time t_(G) in exchange for robustness.

Previously proposed laser-free {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) geometric phase gates (see, e.g., Ospelkaus 2008; Leibfried 2007: Phys. Rev. A 76 032324) included an oscillating magnetic field gradient close to the ions' motional frequency. These gates are appealing because static qubit frequency shifts commute with the gate and can be canceled with a spin-echo sequence (see, e.g., Milburn 2000: Fortschr. Phys. 48 801; Leibfried 2003). However, experimental techniques for generating the gradients usually also result in residual near-resonant electric fields which excite the ion motion and impact gate fidelity (see, Ospelkaus 2008), and these technical challenges limit the implementation of high-fidelity laser-free {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gates.

Previous laser-free trapped-ion quantum logic experiments with oscillating gradients used a pair of near-field microwave gradients, symmetrically detuned about the qubit frequency, to generate the spin-motion coupling used for an entangling gate (see, Ospelkaus 2011; Harty 2016). To reduce off-resonant qubit excitations and ac Zeeman shifts, the microwave magnetic field was minimized at the position of the ions. Recent theoretical work, however, has shown that gates can still be performed in the presence of microwave fields when the qubits are in the dressed state basis with respect to a monochromatic field (see Wolk 2017). In some implementations of geometric phase gates, the microwave field is bichromatic, which complicates analyzing the gate in the dressed-state basis.

As described more fully herein, two-qubit gate dynamics can be derived in the interaction picture with respect to the bichromatic microwave field present in current experimental implementations of geometric phase gates, which is referred to herein as the “bichromatic interaction picture.” As described herein, the dynamics in this interaction picture produce the same final state as in the laboratory frame, as long as the bichromatic fields are turned on and off adiabatically. In certain implementations, the gate basis can be chosen to be either {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) or {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) simply by changing the detuning of the bichromatic field, and it is possible to dynamically decouple from qubit frequency shifts without adding an extra field. In certain implementations, this technique enables {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gates with all fields far detuned from the ions' motional frequencies.

In certain implementations, the intrinsic dynamically decoupled qubit gates described herein can be applied to any trapped-ion entangling gate that is based on microwaves. Certain implementations advantageously provide high-fidelity microwave-based qubit gates (e.g., ZZ gates) that are insensitive to qubit frequency shifts (e.g., caused by magnetic field noise, misalignment of the ions, etc.), that commutes with static errors caused by various miscalibrations, and/or that uses the fields that create the qubit gate to provide dynamical decoupling. Such qubit gates represent an improvement over previously proposed and performed qubit gates which are not intrinsically insensitive to qubit frequency shifts, that utilize added fields to dynamically decouple the qubit gates from noise sources, and/or that utilize precise alignment of the ions and alignment of the gate fields with such external fields.

In certain implementations, the microwave field Rabi frequency magnitude is of the same order as the motional frequencies of the trapped ions. This regime of large microwave field magnitudes (e.g., Ω_(μ)˜ω_(r)) had not previously been investigated or utilized, at least partly because this regime is beyond the peturbative analysis used previously.

In certain implementations, polychromatic gates (e.g., geometric phase gates that comprise multiple simultaneously applied gate fields with optimized amplitudes) advantageously reduce the impact of three types of motional decoherence (e.g., secular frequency shifts, motional heating, and motional dephasing) and have a comparatively smaller trade-off in the gate time t_(G), with less sensitivity to qubit frequency shifts and without using additional control fields.

Gates in the Bichromatic Interaction Picture

Interaction Picture Dynamics

A Hamiltonian Ĥ(t), acting on the state |ψ(t)

, can be written as consisting of two parts: Ĥ(t)=Ĥ _(μ)(t)+Ĥ _(g)(t)  (1) going into the interaction picture with respect to Ĥ_(μ)(t), and Ĥ_(g)(t) is the remainder of the Hamiltonian. Ĥ_(μ)(t) is assumed to commute with itself at all times and no such assumption is made about Ĥ_(g)(t). Transforming into the interaction picture with respect to Ĥ_(μ)(t) gives an interaction picture Hamiltonian Ĥ_(I)(t): Ĥ _(I)(t)=Û ^(†)(t)Ĥ(t)Û(t)+iℏ{circumflex over ({dot over (U)})} ^(†)(t)Û(t)=Û ^(†)(t)Ĥ _(g)(t)Û(t),  (2) where

$\begin{matrix} {{\hat{U}(t)} = {\exp{\left\{ {{- \frac{i}{h}}{\int_{0}^{t}{{dt}^{\prime}{{\overset{\hat{}}{H}}_{\mu}\left( t^{\prime} \right)}}}} \right\}.}}} & (3) \end{matrix}$ In this frame, the time evolution of the transformed state |ϕ(t)

≡Û ^(†)(t)|ψ(t)

,  (4) Is governed by the interaction picture Schrödinger equation iℏ|{dot over (ϕ)}(t)

=Ĥ _(I)(t)|ϕ(t)

.  (5) After applying Ĥ_(I)(t) to |ψ(0)

, for a duration t_(f), the evolution of |ψ(t_(f))

is described by the unitary propagator {circumflex over (T)}_(I)(t_(f)) obtained by solving Eq. (5). Thus, the final state in the original frame is given by: |ψ(t _(f))

=Û(t _(f))|ϕ(t _(f))

=Û(t _(f)){circumflex over (T)} _(I)(t _(f))|ϕ(0)

=Û(t _(f)){circumflex over (T)} _(I)(t _(f))Û ^(†)(0)|ψ(0)

.  (6) If after the time evolution Û_(I)(t_(f))→Î (Û^(†)(0)=Î trivially), where Î is the identity operator, the evolution of |ψ(t_(f))

can be expressed as: |ψ(t _(f))

→{circumflex over (T)} _(I)(t _(f))|ψ(0)

,  (7) which means that the propagator in the interaction picture is equal to the propagator in the original frame. As described herein, for this system, the limit Û_(I)(t_(f))→Î can be realized by turning Ĥ_(μ) on and off adiabatically with pulse shaping.

Microwave-Driven Bichromatic Gates

A general Hamiltonian for microwave-based gates between n trapped ions with identical qubit frequencies (which can be used to describe both laser-based gates and microwave-based gates) can be expressed as:

$\begin{matrix} {{{\overset{\hat{}}{H}}_{lab}(t)} = {{\frac{\hslash\omega_{0}}{2}{\overset{\hat{}}{S}}_{z}} + {{\hslash\omega}_{r}{\overset{\hat{}}{a}}^{\dagger}\overset{\hat{}}{a}} + {2\hslash\Omega_{\mu}{\overset{\hat{}}{S}}_{i}\left\{ {{\cos\left( {\left\lbrack {\omega_{0} + \delta} \right\rbrack t} \right)} + {\cos\left( {\left\lbrack {\omega_{0} - \delta} \right\rbrack t} \right)}} \right\}} + {2{\hslash\Omega}_{g}{f(t)}S_{j}{\left\{ {\overset{\hat{}}{a} + {\overset{\hat{}}{a}}^{\dagger}} \right\}.}}}} & (8) \end{matrix}$ The n-ion Pauli spin operators can be defined as Ŝ_(i)=Σ_(n){circumflex over (σ)}_(i,n), with i∈{x, y} and j∈{x, y, z}, where z refers to the qubit quantization axis and ω₀ is the qubit frequency. In certain implementations, an ion crystal has internal states that are coupled via a motional mode with frequency ω_(r) and creation (annihilation) operators â^(†)(â), with all other motional modes sufficiently detuned from ω_(r) that they will not couple to the spins. The magnetic field gradient Rabi frequency is Ω_(g) and the microwave Rabi frequency is Ω_(μ), with the Ω_(μ) term representing two fields of equal amplitude, detuned from the qubit transition frequency by ±δ (with δ<<ω₀), which only affect the internal states. The Ω_(g) term couples the internal states and the motion and can be implemented with a gradient (e.g., along the motional mode) of the j component of a magnetic field. The time dependence f(t) of the gradient can be an arbitrary function of time (e.g., f(t) can be either constant or sinusoidally oscillating).

Eq. (8) can be transformed into the interaction picture with respect to the “bare” ion Hamiltonian Ĥ₀=ℏω₀Ŝ_(z)/2+ℏω_(r)â^(†)â, and make a rotating wave approximation to eliminate terms near 2ω₀, yielding: Ĥ(t)=Ĥ_(μ)(t)+Ĥ _(g)(t)=2ℏΩ_(μ) Ŝ _(i) cos(δt)+2ℏΩ_(g) f(t)Ŝ _(j) {âe ^(−iω) ^(r) ^(t) +â ^(†) e ^(iω) ^(r) ^(t)}.  (9) This reference frame can be referred to as the ion frame, and the Hamiltonians Ĥ_(μ)(t), and Ĥ_(g)(t) can be referred to as the microwave field term and the gradient term, respectively, which are the transformed third and fourth terms of Eq. (8). Eq. (9) is valid unconditionally for j=z. In the case that j∈{x, y}, Eq. (9) holds as long as the gradient has a bichromatic oscillating time dependence in the lab frame as described in Eq. (8), i.e., f(t)=cos([ω₀+δ′]t)+cos([ω₀−δ′]t) for some δ′<<ω₀. After transforming into the ion frame and dropping fast-rotating terms near 2ω₀, f(t) becomes cos(δ′t) in Eq. (9). This form of bichromatic oscillating gradient is used herein in describing the near-qubit-frequency oscillating gradient case, with δ′=δ, since the microwave and gradient terms originate from the same field.

In Eqs. (8) and (9), the operator Ŝ_(j) in the gradient term also implicitly incorporates information about the motional mode, and is defined here to correspond to a center-of-mass mode (for two identical ions it could be trivially extended to an out-of-phase mode by setting Ŝ_(j)≡∂_(j,1)−∂_(j,2)). For simplicity, all ions are assumed to be the same, and can be addressed with a single pair of microwave fields. The formalism used herein can be generalized to the case of multiple qubit frequencies—either for multiple ion species, or for ions of the same species (see, e.g., Khromova 2012)—by using multiple pairs of microwave fields.

In certain implementations, the gates are described using a reference frame having the ion frame Hamiltonian from Eq. (9) in the bichromatic interaction picture with respect to the microwave field term Ĥ_(μ)(t). This reference frame, rotating at a nonuniform rate, has been utilized in the context of laser-driven gates (see, Roos 2008) to accurately quantify the effect of an off-resonant field. In certain implementations, the Ω_(μ) term is constant (e.g., neglecting pulse shaping), while in certain other implementations, adiabatic pulse shaping is used as described herein.

The bichromatic interaction picture can be expressed using the transformation:

$\begin{matrix} {{\hat{U}(t)} = {{\exp\left\{ {{- \frac{i}{\hslash}}{\int_{0}^{t}{{dt}^{\prime}{H_{\mu}\left( t^{\prime} \right)}}}} \right\}} = {{\exp\left\{ {{- 2}i\Omega_{\mu}{\overset{\hat{}}{S}}_{i}{\int_{0}^{t}{{dt}^{\prime}{\cos\left( {\delta t^{\prime}} \right)}}}} \right\}} = {\exp{\left\{ {{- i}{F(t)}{\overset{\hat{}}{S}}_{i}} \right\}.}}}}} & (10) \end{matrix}$ where F(t)≡[2Ω_(μ) sin(δt)]/δ. The interaction picture Hamiltonian is then: Ĥ _(I)(t)=2ℏΩ_(g) f(t){âe ^(−iω) ^(r) ^(t) +â ^(†) e ^(iω) ^(r) ^(t) }e ^(iF(t)Ŝ) ^(i) Ŝ _(j) e ^(iF(t)Ŝ) ^(i) .  (11) Focusing on the Pauli operators in Eq. (11): e ^(iF(t)Ŝ) ^(i) Ŝ _(j) e ^(−iF(t)Ŝ) ^(i) ={Î cos(F(t))+iŜ _(i) sin(F(t))}Ŝ _(j) e ^(−iF(t)Ŝ) ^(i) =Ŝ _(j) +i[Ŝ _(i) ,Ŝ _(j)] sin(F(t))e ^(−iF(t)Ŝ) ^(i) .  (12) Inserting this into Eq. (11) gives: Ĥ _(I)(t)=2ℏΩ_(g) f(t){âe ^(−iω) ^(r) ^(t) +â ^(†) e ^(iω) ^(r) ^(t) }{Ŝ _(j) +i[Ŝ_(i) ,Ŝ _(j)]sin(F(t))e ^(−iF(t)Ŝ) ^(i) }.  (13) If i=j, then Eq. (12)→Ŝ_(j), and Ĥ_(I)(t) is equal to Ĥ_(g)(t). However, if i≠j, then Eq. (13) becomes: Ĥ _(I)(t)=2ℏΩ_(g) f(t){âe ^(−iω) ^(r) ^(t) +â ^(†) e ^(iω) ^(r) ^(t) }{Ŝ _(j) cos(2F(t))−ϵ_(ijk) Ŝ _(k) sin(2F(t))}.  (14)

Using the Jacobi-Anger expansion (see, e.g., Abramowitz 1972: Handbook of mathematical functions (New York: Dover)), the bichromatic interaction Hamiltonian can be expressed as:

$\begin{matrix} {{{{\overset{\hat{}}{H}}_{I}(t)} = {2\hslash\Omega_{g}{f(t)}\left\{ {{\overset{\hat{}}{a}e^{{- i}\omega_{r}t}} + {{\overset{\hat{}}{a}}^{\dagger}e^{i\omega_{r}t}}} \right\}\left\{ {{{\overset{\hat{}}{S}}_{j}\left\lbrack {{J_{0}\left( \frac{4\Omega_{\mu}}{\delta} \right)}2{\sum\limits_{n = 1}^{\infty}{{J_{2n}\left( \frac{4\Omega_{\mu}}{\delta} \right)}{\cos\left( {2n\delta t} \right)}}}} \right\rbrack} - {2\epsilon_{ijk}{\hat{S}}_{k}{\sum\limits_{n = 1}^{\infty}{{J_{{2n} - 1}\left( \frac{4\Omega_{\mu}}{\delta} \right)}{\sin\left( {\left\lbrack {{2n} - 1} \right\rbrack\delta t} \right)}}}}} \right\}}},} & (15) \end{matrix}$ where J_(n) is the nth Bessel function, and ϵ_(ijk) is the Levi-Civita symbol. Two possible functional forms of the time dependence f(t) of the gradient of the j component of the magnetic field can be considered: (i) sinusoidal, which corresponds to the oscillating magnetic field gradient from an ac-current-carrying wire (see, Ospelkaus 2011; Harty 2016), or (ii) constant, which corresponds to a magnetic field gradient induced by a permanent magnet (see, Khromova 2012, Lake 2015; Weidt 2016: Phys. Rev. Lett. 117 220501) or a dc-current-carrying wire (see, Welzel 2018: arXiv preprint arXiv:1801.03391).

When i≠j, Eq. (15) shows an infinite series of resonances in the bichromatic interaction picture, each with a strength proportional to a Bessel function. In certain implementations, specific values of the motional mode frequency ω_(r), the detuning δ of the magnetic field gradient Rabi frequency Ω_(g) from the qubit transition frequency ω₀, and n can be chosen with a given f such that one of these terms in Eq. (15) is resonant (e.g., stationary or slowly varying in time).

For example, in certain implementations, the detuning δ is large compared to the magnetic field gradient Rabi frequency Ω_(g), such that δ>>Ω_(g)f(t) (see, Ospelkaus 2008; Ospelkaus 2011; Harty 2016) and near any particular resonance, the off-resonant terms in Eq. (15), whose effect scales as (Ω_(g)f(t)/δ)², can be ignored. In the case of multiple qubit transition frequencies, there can be additional terms in Eq. (15) at other frequencies. Whether or not these terms can be neglected can depend on the specific values of the qubit transition frequencies as well as the detuning δ, the magnetic field gradient Rabi frequency Ω_(g), and the time dependence f(t).

Even Bessel function resonances (e.g., J₀, J₂, J₄, . . . ) correspond to gate operations where the spin operator Ŝ_(j) for the gate is identical to the spin operator for the gradient term in Eq. (9). The odd Bessel function resonances (e.g., J₁, J₃, J₅, . . . ) correspond to gates whose spin operators Ŝ_(k) are orthogonal to both the microwave and gradient spin operators Ŝ_(i) and Ŝ_(j), respectively. For example, where i∈{x,y} and j=z, the even and odd Bessel function resonances correspond to {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) and {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gates (specifically, {circumflex over (σ)}_(x)⊗{circumflex over (σ)}_(x) or {circumflex over (σ)}_(y)⊗{circumflex over (σ)}_(y) gates, depending on the choice of i).

FIG. 1 shows the relative strengths of the gate Rabi frequencies corresponding to the first three resonances versus 4Ω_(μ)/δ in accordance with certain implementations described herein. These strengths shown in FIG. 1 are plotted in the bichromatic interaction picture in which the microwave field term (corresponding to the microwave spin operator Ŝ_(i)) does not commute with the gradient field term (corresponding to the gradient spin operator Ŝ_(j)) in the Hamiltonian of Eq. (15).

In certain implementations, a dynamically decoupled entangling gate operation can be performed without adding an extra field, which can simplify the physical configuration as compared to other configurations in which an additional field is added. In general, dynamical decoupling (see, e.g., Viola 1998; Viola 1999; Uhrig 2007: Phys. Rev. Lett. 98 100504) is a useful tool for error suppression in trapped-ion quantum logic experiments (see, e.g., Harty 2016; Bermudez 2012; Tan 2013; Biercuk 2009: Phys. Rev. A 79 062324; Timoney 2011: Nature 476 185; Piltz 2013: Phys. Rev. Lett. 110 200501; Manovitz 2017: Phys. Rev. Lett. 119 220505). For example, by adding an oscillating magnetic field at the qubit transition frequency that commutes with the gate but not with qubit frequency fluctuations, thus suppressing the leading source of error while leaving the gate unaffected, Harty 2016 disclosed continuous dynamical decoupling that achieved an entangling gate fidelity of approximately 0.997, making the gate operation highly insensitive to qubit frequency fluctuations.

The dynamical decoupling of certain implementations described herein can be illustrated by adding an error term to the Hamiltonian shown in Eq. (9):

$\begin{matrix} {{{\hat{H}}_{z} = {\frac{\hslash\varepsilon}{2}{\overset{\hat{}}{S}}_{z}}},} & (16) \end{matrix}$ where ε is a qubit frequency shift (which can be time-dependent), arising, for example from environmental noise, control field fluctuations, and/or miscalibration of the qubit transition frequency ω₀. With i∈{x, y}, this term can be transformed into the bichromatic interaction picture:

$\begin{matrix} {{\hat{H}}_{I,z} = {\frac{\hslash\varepsilon}{2}{\left\{ {{{\overset{\hat{}}{S}}_{z}\left\lbrack {{J_{0}\left( \frac{4\Omega_{\mu}}{\delta} \right)} + {2{\sum\limits_{n = 1}^{\infty}{{J_{2n}\left( \frac{4\Omega_{\mu}}{\delta} \right)}{\cos\left( {2n\delta t} \right)}}}}} \right\rbrack} + {2\epsilon_{ijk}{\overset{\hat{}}{S}}_{k}{\sum\limits_{n = 1}^{\infty}{{J_{{2n} - 1}\left( \frac{4\Omega_{\mu}}{\delta} \right)}{\sin\left( {\left\lbrack {{2n} - 1} \right\rbrack\delta t} \right)}}}}} \right\}.}}} & (17) \end{matrix}$

If the qubit frequency shift E varies slowly (e.g., on timescales of 1/δ), then the only term Ĥ_(I,z) that is not oscillating near a multiple of δ is proportional to J₀(4Ω_(μ)/δ). Therefore, in certain implementations in which (4Ω_(μ)/δ) is approximately equal to the first zero of the J₀ Bessel function (e.g., 2.405), intrinsic dynamical decoupling is achieved since only the fast-oscillating qubit frequency shift terms are left, and these terms contribute negligible dephasing (e.g., contributions scaling as (ε/δ)²). As can be seen in FIG. 1 , the value of (4Ω_(μ)/δ) where intrinsic dynamical decoupling is achieved occurs near the maxima of the J₁ and J₂ Bessel functions, so operating at this value of (4Ω_(μ)/δ) only results in a relatively small reduction in gate speed (e.g., reduction of about 11% relative to the fastest achievable J₁ and J₂ gates).

In certain implementations, the microwave bichromatic field is smoothly ramped on and off over a time τ>>2π/δ, such that the final wave function in the ion frame approaches the final wave function in the bichromatic interaction picture (e.g., adiabatic pulse shaping). By turning the microwave pair (as parameterized by Ω_(μ)) is turned on and off in such a manner that Û(t_(i))=Î and Û(t_(f))=Î, the state evolution given by the propagator {circumflex over (T)}_(I)(t_(i), t_(f)) of Ĥ_(I)(t) applies to both the lab frame and the transformed basis in the bichromatic interaction picture. In other words, the unitary transformation defined by Eq. (3) approaches the identity operator, Û(t_(f))→Î. In certain implementations, this condition can be achieved by turning the microwave pair on and off slowly with respect to 1/δ, or by choosing the gate time t_(G) (e.g., the time during which the microwave pair is on) such that t_(G)δ is an integer multiple of 2π.

In certain implementations, such microwave pulse shaping can be modeled by modifying the microwave field term in Eq. (9) to include a time-dependent envelope g(t) with a continuous first derivative: Ĥ _(μ)(t)→2Ω_(μ) g(t)cos(δt)Ŝ _(i),  (18) where g(t) varies slowly (e.g., on the timescale 2π/δ) and the microwave Rabi frequency is equal to zero at the beginning and end of the gate operation, and is constant in between the ramps, such that the pulse shape satisfied the following: g(t=0,t _(f))=0 g(τ≤t≤t _(f)−τ)=1,  (19) where t_(f) is the final gate time. At the end of the gate operation, the unitary transformation into the bichromatic interaction picture can be expressed as: Û(t _(f))=exp{−i∫ ₀ ^(t) ^(f) dt′2Ω_(μ) g(t′)cos(δt′)Ŝ _(i)}.  (20) and, by integrating by parts:

$\begin{matrix} {{\hat{U}\left( t_{f} \right)} = {\exp{\left\{ {\frac{2i\Omega_{\mu}}{\delta}\left( {{\int_{0}^{\tau}{{dt}^{\prime}{\overset{.}{g}\left( t^{\prime} \right)}{\sin\left( {\delta t^{\prime}} \right)}}} + {\int_{t_{f - \tau}}^{t_{f}}\ {{dt}^{\prime}{g\left( t^{\prime} \right)}{\sin\left( {\delta t^{\prime}} \right)}}}} \right){\overset{\hat{}}{S}}_{i}} \right\}.}}} & (21) \end{matrix}$

In certain implementations in which ġ(t) is a slowly varying function with respect to sin(δt), the larger the value of τ is relative to 2π/δ, the smaller the values of the two integrals are in Eq. (21). Thus, in the limit τ>>2π/δ, Û(t_(f))→Î, and the final ion frame state approaches the final interaction picture state. This effect is independent of the actual shape of the pulse envelope, provided it is slowly varying, as described more fully herein with regard to the example of the near-motional-frequency oscillating gradient. In certain implementations, pulse shaping can slightly change the optimal gate times due to the changing Rabi frequency during the rise and fall times.

EXAMPLE IMPLEMENTATIONS

Various implementations of microwave-based entangling gates, with corresponding Pauli spin operators Ŝ_(i), Ŝ_(j), and time dependence f(t) of the magnetic field gradient, can be described using the formalism herein.

Example 1: Static Magnetic Field Gradient

In a first example of a gate utilizing a microwave spin-motion coupling scheme, the magnetic field gradient is static (e.g., f(t) is constant, as can be provided by a permanent magnet) and is used in combination with one or more microwave fields (see, e.g., Minert 2001; Lake 2015). The ion frame Hamiltonian can be expressed as: Ĥ(t)=2ℏΩ_(μ) Ŝ _(x) cos(δt)+2ℏΩ_(g) Ŝ _(z) {âe ^(−iω) ^(r) ^(t) +â ^(†) e ^(iω) ^(r) ^(t)}.  (22) which corresponds to Ŝ_(i)=Ŝ_(x), Ŝ_(j)=Ŝ_(z), and f(t)=1 in Eq. (9), and Eq. (15) becomes:

$\begin{matrix} {{{\overset{\hat{}}{H}}_{I}(t)} = {2\hslash\Omega_{g}\left\{ {{\overset{\hat{}}{a}e^{{- i}\omega_{r}t}} + {{\overset{\hat{}}{a}}^{\dagger}e^{i\omega_{r}t}}} \right\}{\left\{ {{{\overset{\hat{}}{S}}_{z}\left\lbrack {{J_{0}\left( \frac{4\Omega_{\mu}}{\delta} \right)} + {2{\sum\limits_{n = 1}^{\infty}{{J_{2n}\left( \frac{4\Omega_{\mu}}{\delta} \right)}{\cos\left( {2n\delta t} \right)}}}}} \right\rbrack} + {2{\hat{S}}_{y}{\sum\limits_{n = 1}^{\infty}{{J_{{2n} - 1}\left( \frac{4\Omega_{\mu}}{\delta} \right)}{\sin\left( {\left\lbrack {{2n} - 1} \right\rbrack\delta t} \right)}}}}} \right\}.}}} & (23) \end{matrix}$ By having Ω_(g)<<δ (e.g., keeping only the near resonant terms in Eq. (23)) provides a {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate (e.g., a “ZZ gate”) when 2nδ is approximately equal to the motional mode frequency ω_(r), and provides a {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate (e.g., a {circumflex over (σ)}_(y)⊗{circumflex over (σ)}_(y) gate or “YY gate”) when (2n−1)δ is approximately equal to the motional mode frequency ω_(r).

Example 2: Near-Qubit-Frequency Oscillating Gradient

In a second example of a gate utilizing a microwave spin-motion coupling scheme, the near-field gradient oscillates at a frequency close to the qubit frequency ω₀ (see, e.g., Ospelkaus 2008; Ospelkaus 2011). The gradient and the microwave terms are caused by the same field and point in the same direction. The ion frame Hamiltonian can be expressed as: Ĥ(t)=2ℏΩ_(μ) cos(δt)Ŝ _(x)+2ℏΩ_(g) cos(δt)Ŝ _(x) {âe ^(−iω) ^(r) ^(t) +â ^(†) e ^(iω) ^(r) ^(t)}.  (24) which corresponds to Ŝ_(i)=Ŝ_(x), Ŝ_(j)=Ŝ_(x), and f(t)=cos(δt) in Eq. (9). Since the microwave term commutes with the gradient term, the bichromatic interaction picture Hamiltonian can be expressed as: Ĥ _(I)(t)=2ℏΩ_(g) cos(δt)Ŝ _(x) {âe ^(−iω) ^(r) ^(t) +â ^(†) e ^(iω) ^(r) ^(t)}.  (25) This Hamiltonian provides a {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate (e.g., a {circumflex over (σ)}_(x)⊗{circumflex over (σ)}_(x) gate or “XX gate”). The infinite series of resonances in Eq. (15) is absent because the microwave term and the gradient term commute (i=j). In the presence of a qubit frequency shift of the form in Eq. (16), transforming into the bichromatic interaction picture adds a term to Eq. (25) of the form shown in Eq. (17), which can be analyzed as disclosed herein regarding intrinsic dynamical decoupling.

Example 3: Near-Motional-Frequency Oscillating Gradient

In a third example of a gate utilizing a microwave spin-motion coupling scheme, separate gradient and microwave fields can be used that are oscillating at near-motional and near-qubit frequencies, respectively. A previous laser-based configuration utilized a running optical lattice to create an oscillating gradient of the ac Stark shift near the ion motional frequencies (Ding 2014: Phys. Rev. Lett. 113 073002). In certain implementations of a microwave-based configuration, separate near-qubit and near-motional frequency currents can be superimposed on near-field electrodes in a surface electrode trap (see, Srinivas 2019). In certain such implementations in which the gradient lies along the quantization axis and the microwave fields are perpendicular to the quantization axis, the ion frame Hamiltonian can be expressed as: Ĥ(t)=2ℏΩ_(μ) cos(δt)Ŝ _(x)+2ℏΩ_(g) cos(ω_(g) t)Ŝ _(z) {âe ^(−ω) ^(r) ^(t) +â ^(†) e ^(iω) ^(r) ^(t)},  (26) where ω_(g) is the frequency of the oscillating gradient field, Ŝ_(i)=Ŝ_(x), Ŝ_(j)=Ŝ_(z), and f(t)=cos(ω_(g)t) so that Eq. (15) becomes:

$\begin{matrix} {{{\overset{\hat{}}{H}}_{I}(t)} = {2\hslash\Omega_{g}{\cos\left( {\omega_{g}t} \right)}\left\{ {{\overset{\hat{}}{a}e^{{- i}\omega_{r}t}} + {{\overset{\hat{}}{a}}^{\dagger}e^{i\omega_{r}t}}} \right\}{\left\{ {{{\overset{\hat{}}{S}}_{z}\left\lbrack {{J_{0}\left( \frac{4\Omega_{\mu}}{\delta} \right)} + {2{\sum\limits_{n = 1}^{\infty}{{J_{2n}\left( \frac{4\Omega_{\mu}}{\delta} \right)}{\cos\left( {2n\delta t} \right)}}}}} \right\rbrack} + {2{\hat{S}}_{y}{\sum\limits_{n = 1}^{\infty}{{J_{{2n} - 1}\left( \frac{4\Omega_{\mu}}{\delta} \right)}{\sin\left( {\left\lbrack {{2n} - 1} \right\rbrack\delta t} \right)}}}}} \right\}.}}} & (27) \end{matrix}$

The near-motional-frequency oscillating gradient example of Eq. (27) is similar to the static field example of Eq. (23), only with resonances occurring when δ is close to (e.g., within 10%) of an integer multiple of |ω_(r)±ω_(g)|, rather than ω_(r). As a result, the Bessel function extrema and roots can be reached with lower values of Ω_(μ) than for the static or near-qubit frequency gradient examples. In certain implementations, this third example can produce a {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate (e.g., a {circumflex over (σ)}_(y)⊗{circumflex over (σ)}_(y) gate or “YY gate”), while in certain other implementations, this third example can produce a {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate (e.g., “ZZ gate”).

YY Gate of Example 3

In certain implementations, the near-motional-frequency oscillating gradient can be used in which a pair of microwave fields, oscillating near the qubit transition frequency ω₀ and polarized in the {circumflex over (x)} direction, are combined with a gradient field oscillating near the motional frequency ω_(r) and polarized in the {circumflex over (z)} direction. However, the qualitative results described herein apply to the other two examples described herein. By setting δ˜(ω_(r)−ω_(g)) and only keeping the resonant terms in Eq. (27) gives:

$\begin{matrix} {{{{\overset{\hat{}}{H}}_{I}(t)} \simeq {i\hslash\Omega_{g}{J_{1}\left( \frac{4\Omega_{\mu}}{\delta} \right)}{\overset{\hat{}}{S}}_{y}\left\{ {{{\overset{\hat{}}{a}}^{\dagger}e^{{- i}\Delta t}} - {\overset{\hat{}}{a}e^{i\Delta t}}} \right\}}},} & (28) \end{matrix}$ where Δ≡δ−(ω_(r)−ω_(g)). Eq. (28) corresponds to a {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate (e.g., a {circumflex over (σ)}_(y)⊗{circumflex over (σ)}_(y) gate or “YY gate”) with a Rabi frequency of Ω_(ϕ)≡Ω_(g)J₁(4Ω_(μ)/δ). While the time propagator for the ion frame Hamiltonian (see, Eq. (26)) is fairly complicated to solve analytically, the time propagator for this interaction picture Hamiltonian is well-known (see, e.g., Mølmer 1999; Sørensen 2000; Roos 2008; Solano 1999: Phys. Rev. A 59 R2539). At t_(f)=2π/Δ, the propagator is:

$\begin{matrix} {{{\overset{\hat{}}{T}}_{I}\left( t_{f} \right)} = {\exp{\left\{ {{- \frac{2\pi i}{\Delta^{2}}}\left( {\Omega_{\varnothing}{\overset{\hat{}}{S}}_{y}} \right)^{2}} \right\}.}}} & (29) \end{matrix}$ For a system starting in the ground state |↓↓

, this gate generates a maximally entangled Bell state when Δ=4Ω_(φ):

$\begin{matrix} {{\left. ❘{Bell} \right\rangle \equiv {\frac{1}{\sqrt{2}}\left\{ {\left. ❘\left. \downarrow\downarrow \right. \right\rangle + {i\left. ❘\left. \uparrow\uparrow \right. \right\rangle}} \right\}}},} & (30) \end{matrix}$ ignoring an overall phase.

FIGS. 2A and 2B show numerical simulations of the fidelity

≡

Bell|{circumflex over (ρ)}(t)|Bell

for the {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate of the maximally entangled Bell state of Eq. (30) versus time (normalized to t_(f)), where {circumflex over (ρ)}(t) is the reduced density operator for the qubit subspace, in accordance with certain implementations described herein. Both FIGS. 2A and 2B simulate the {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate operation for a two-ion system, with the high frequency thin line corresponding to the full ion frame Hamiltonian of Eq. (26), and the thick generally horizontal line corresponding to the bichromatic interaction picture Hamiltonian of Eq. (28). The physical parameters used in FIGS. 2A and 2B were Ω_(μ)/2π=500 kHz, Ω_(g)/2π=1 kHz, ω_(r)/2π=6.5 MHz, and ω_(g)/2π=5 MHz, and Ŝ_(i)=Ŝ_(x), Ŝ_(j)=Ŝ_(z).

FIG. 2A shows the gate fidelity

, in both the bichromatic interaction picture and the ion frame, for the {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate operation with no pulse shaping, where large-amplitude oscillations at δ make the ion frame gate fidelity highly sensitive to the exact value of t_(f). In the interaction picture (e.g., |ϕ(t)

as opposed to |ψ(t)

as disclosed herein with regard to the interaction picture dynamics), the state |Bell

is created with

=1. However, in the ion frame, the fidelity

oscillates according to

∝ cos⁴((2Ω_(μ)/δ)sin(δt)). The peak value of the ion frame fidelity

agrees with the fidelity

from the bichromatic interaction picture to within the numerical accuracy of the simulations (approximately 10⁻⁵), indicating that the off-resonant terms dropped from Eq. (27) do not impact fidelity

at this level.

FIG. 2B shows the same {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate operation including a microwave envelope g(t) with a 10 μs Blackman edge at the rising and falling edges (e.g., at the beginning and the end of the gate sequence). As seen in FIG. 2B, the ion frame fidelity smoothly converges to the interaction picture fidelity at the end of the gate, as disclosed herein with regard to the adiabatic pulse shaping. Thus, even in the presence of a strong bichromatic microwave field term, high fidelity gates can be implemented. Certain such implementations enable simplification of the physical configuration, since the microwave magnetic field does not have to be minimized at the ions' positions. Furthermore, in certain implementations, the strength of the microwave magnetic field can be tuned to decouple the system from qubit frequency shifts without additional drive fields.

FIG. 3A shows the effect of the intrinsic dynamical decoupling on the {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate in accordance with certain implementations described herein. FIG. 3A is a plot of the gate fidelity

of the gate operation creating the maximally entangled Bell state of Eq. (30) as a function of the normalized static qubit frequency shift ϵ/Ω_(g). The plots of FIG. 3A were calculated using the physical parameters of FIGS. 2A and 2B, but varying the microwave Rabi frequency Ω_(μ) to change the arguments of the Bessel functions and using numerical integration of the full ion frame Hamiltonian of Eq. (26). As seen in FIG. 3A, for most values of Ω_(μ), the gate fidelity

is highly sensitive to qubit frequency fluctuations. For both a gate of FIGS. 2A and 2B (e.g., 4Ω_(μ)/δ≃1.333), and for a gate with Ω_(μ) increased to achieve a maximized gate speed (e.g., 4Ω_(μ)/δ≃1.841), the value of the gate fidelity

, when |ϵ|/Ω_(g)≥≳1, drops to {tilde over ( )}0.5. However, when Ω_(μ) is further increased such that 4Ω_(μ)/δ≃2.405 (i.e., the first root of the J₀ Bessel function), the gate fidelity

becomes significantly less sensitive to ϵ, giving

≥0.95 for |ϵ|/Ω_(g)≤5.

FIG. 3B is a plot of the gate infidelity 1−

of the {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate as a function of the frequency ω_(ε) at which the qubit shift ε is time-varying (e.g., ε=ε₀ cos(ω_(ε)t) with the value of ε₀=Ω_(g)) in accordance with certain implementations described herein. This value of ε₀ corresponds to a significantly larger qubit shift than the value |ε₀|<<Ω_(g) typically used experimentally (see, Harty 2016). FIG. 3B includes the gate infidelity 1−

for an intrinsically dynamically decoupled {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate (e.g., 4Ω_(μ)/δ≃2.405; solid black line) and for a {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate with Ω_(μ) increased to achieve a maximized gate speed (e.g., 4Ω_(μ)/δ≃1.841; dashed line). As shown in FIG. 3B, intrinsic dynamical decoupling in certain implementations protects against qubit energy shifts at frequencies up to ≈10Ω_(g). FIG. 3B also shows the sensitivity of intrinsic dynamical decoupling to small fluctuations in Ω_(μ)/δ by including the infidelity when the ratio 4Ω_(μ)/δ is shifted by 1% above and below the intrinsic dynamical decoupling point (e.g., 4Ω_(μ)/δ≃2.405±1%; grey solid lines). Similar simulations for various values of ε₀ can show that the infidelity scales as (ε₀/Ω_(g))² for ε₀≤Ω_(g).

In contrast with the dynamically decoupled {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate disclosed by Harty 2016, certain implementations described herein do not utilize an additional field to provide the dynamical decoupling, and the microwave field term generating the dynamical decoupling does not have to commute with the gradient term in the Hamiltonian. In certain implementations, as described more fully herein, the infinite series of resonances resulting from the microwave field term not commuting with the gradient advantageously provides a {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) microwave gate in which all frequencies are detuned from the ions' motional modes.

ZZ Gate of Example 3

In certain implementations, as described herein, dynamical decoupling can be beneficial for high-fidelity {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gates, because the terms in the Hamiltonian that represent qubit frequency shifts do not commute with the {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate. However, qubit frequency shifts do commute with a {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate, so, in certain implementations described herein, a simple spin-echo sequence can substantially cancel (e.g., completely cancel) the effect of static qubit frequency shifts. Previously, the only proposed technique for performing microwave-based {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gates included generating oscillating gradients near the ions' motional frequencies, where experimental imperfections can give rise to electric fields that excite the ions' motion and can reduce fidelity, making the {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate difficult to perform in practice (see, Ospelkaus 2008).

Referring back to Eq. (27), a {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate having a near-motional-frequency oscillating gradient is provided when 2nδ˜|ω_(r)−ω_(g)|. For n=0, Eq. (27) corresponds to a {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate with ω_(g)≃ω_(r), and for n=1, Eq. (27) can be expressed as:

$\begin{matrix} {{{{\overset{\hat{}}{H}}_{I}(t)} \simeq {{\hslash\Omega}_{g}{J_{2}\left( \frac{4\Omega_{\mu}}{\delta} \right)}{\overset{\hat{}}{S}}_{z}\left\{ {{\overset{\hat{}}{a}e^{i\Delta t}} - {{\overset{\hat{}}{a}}^{\dagger}e^{{- i}\Delta t}}} \right\}}},} & (31) \end{matrix}$ where Δ≡2δ−(ω_(r)−ω_(g)). Eq. (31) corresponds to a {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate (e.g., a “ZZ gate”) in which both the frequency ω_(g) of the oscillating gradient field and the detuning δ of the microwave Rabi frequency Ω_(μ) from the qubit transition frequency ω₀ can deviate significantly from the motional mode frequency ω_(r), which relaxes the constraints on residual electric fields. These electric fields can drive the ion motion off-resonantly at ω_(g), and for a given electric field strength, the resulting motional amplitude can scale as 1/(ω_(r) ²−ω_(g) ²).

In certain embodiments, the {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate operation is performed with a spin-echo pulse after the first of two loops in phase space, thereby substantially canceling (e.g., completely canceling) the effect of the static shifts. In certain implementations, intrinsic dynamical decoupling can be applied to the {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate, in addition application of the spin echo.

FIG. 3A shows the substantial (e.g., complete) insensitivity to static qubit shifts for the {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate operation (dotted line). FIG. 4 is a plot of the gate infidelity 1−

of the {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate as a function of the frequency ω_(ε) at which the qubit shift ε is time-varying (e.g., ε=ε₀ cos(ω_(ε)t) with the value of ε₀=Ω_(g)) in accordance with certain implementations described herein. This value of ε₀ corresponds to a significantly larger qubit shift than the value |ε₀|<<Ω_(g) as typically used experimentally (see, Harty 2016). FIG. 4 includes the gate infidelity 1−

for an intrinsically dynamically decoupled {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate (e.g., 4Ω_(μ)/δ≃2.405; solid black line) and for a {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate with Ω_(μ) increased to achieve a maximized gate speed J₂(4Ω_(μ)/δ≃3.054)≃0.49 (dashed line). FIG. 4 also shows the sensitivity of intrinsic dynamical decoupling to small fluctuations in Ω_(μ)/δ by including the infidelity when the ratio 4Ω_(μ)/δ is shifted by 1% above and below the intrinsic dynamical decoupling point (e.g., 4Ω_(μ)/δ≃2.405±1%; grey solid lines), showing the sensitivity of the {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate to time-varying qubit shifts. While the {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate is less sensitive to static (ω_(ε)=0) noise, it remains sensitive to noise with larger values of ω_(g). Similar simulations for various values of ε₀ can show that the infidelity scales as (ε₀/Ω_(g))² for ε₀≤Ω_(g).

Further Implementations of an Example ZZ Gate

In certain implementations, the ZZ gate is configured to provide simultaneous robustness to qubit frequency shifts and to motional decoherence without utilizing additional control fields (e.g., while utilizing only two microwave magnetic fields and one near-motional-frequency magnetic field gradient to perform the gate operation). Certain such implementations advantageously provide a combination of increased gate fidelity for laser-free entangling gates without reducing the gate speed and with decreased experimental overhead (e.g., reduced complexity of the driving fields). For example, a laser-free polychromatic gate system disclosed by Webb 2018 utilized two bichromatic microwave pairs close to the frequency of the side band of a static gradient (ω_(g)=0) and a total of 12 oscillating control fields plus a strong static magnetic field gradient, certain implementations described herein only utilize three oscillating control fields that include a strong oscillating magnetic field gradient. In addition, while the system disclosed by Webb 2018 produces unavoidable errors unless Ω_(μ)/ω_(r)<<1 (which limits the achievable gate speed), certain implementations described herein do not have such a limitation on Ω_(μ).

FIG. 5A is a plot of the frequency spectrum of two microwave fields (frequencies: ω₀+δ and ω₀−δ), a magnetic field gradient field (frequency: ω_(g)), and a frequency ω_(r) of a shared collective motional mode, and a qubit transition frequency ω₀, in accordance with certain implementations described herein. The two microwave frequency fields are symmetrically detuned around the qubit transition frequency ω₀ by ±δ, and a separate magnetic field gradient field oscillates at ω_(g). As shown in FIG. 5A, ω_(g) is approximately one-third of the frequency ω_(r) of the shared collective motional mode. Neglecting noise, the dynamics of the example ZZ gate are governed by the Hamiltonian of Eq. (26), where Ω_(g) is the magnetic field gradient Rabi frequency, Ω_(μ) is the microwave Rabi frequency (e.g., with Ω_(μ)˜ω_(r)), â(â^(†)) is a phonon annihilation (creation) operator, Ŝ_(γ∈{x,y,z}) is a two-ion Pauli spin operator (Ŝ_(γ)≡{circumflex over (σ)}_(γ,1)+{circumflex over (σ)}_(γ,2)), z is taken to be the qubit quantization axis, and δ is the detuning of the microwaves from the qubit transition frequency ω₀. In certain implementations, fast-rotating terms are neglected, and the strength of Ω_(g)/2π is in the kHz regime, and Ω_(μ)/2π can be in the MHz regime.

In certain implementations, the oscillating gradient at ω_(g), in combination with the detuned microwaves, gives rise to two spin-motion-coupling sideband interactions, occurring when δ=ω_(r)±ω_(g), respectively, and the bichromatic microwave pair combines to give an effective modulation of Ω_(μ). In certain implementations, the ZZ gate utilizes a magnetic-field-sensitive qubit transition instead of magnetic-field-insensitive “clock” transitions as are typically used for their long coherence times. Certain implementations comprise storing quantum information in a clock qubit and transferring the state populations to a field-sensitive qubit only during times when an entangling gate is being carried out. Alternatively, in certain other implementations, microwave fields can be applied to field-sensitive transitions to create dressed-state clock qubits (see, e.g., Timoney 2011).

As described herein, the interaction picture Hamiltonian can be expressed as Eq. (27). By setting the conditions: 4δ=(ω_(r)−ω_(g))−jΔ 8δ=(ω_(r)+ω_(g))−(j+1)Δ,  (32) where j is an integer, and Δ=2π/t_(G) is on the order of Ω_(g), the terms □ J₄ and □ J₈ become slowly varying in time with respect to all other terms in the sum. These terms therefore make the dominant contribution to the gate system dynamics, while the other terms that appear in Eq. (27) are significantly off-resonant, with contributions scaling as (Ω_(g)/δ)² (note that Ω_(g)<<δ). In the laboratory frame, the conditions in Eqs. (32) are equivalent to setting δ and ω_(g) to drive both the ω_(r)−ω_(g) and ω_(r)+ω_(g) sidebands simultaneously, as shown in FIG. 1C.

Keeping only these near-resonant terms gives:

$\begin{matrix} {{{{\overset{\hat{}}{H}}_{I}(t)} \simeq {\hslash\Omega_{g}{\overset{\hat{}}{S}}_{z}\left\{ {{{J_{4}\left( \frac{4\Omega_{\mu}}{\delta} \right)}\ \left( {{\overset{\hat{}}{a}e^{{- i}j\Delta t}} + {{\overset{\hat{}}{a}}^{\dagger}e^{ij\Delta t}}} \right)} + {{J_{8}\left( \frac{4\Omega_{\mu}}{\delta} \right)}\left( {{\overset{\hat{}}{a}e^{{- {i({j + 1})}}\Delta t}} + {{\overset{\hat{}}{a}}^{\dagger}e^{{i({j + 1})}\Delta t}}} \right)}} \right\}}},} & (33) \end{matrix}$ which resembles, but differs from, other disclosed forms of a motionally robust polychromatic gates (see, e.g., Webb 2018; Haddadfarshi 2016: New J. Phys. 18, 123007; Shapira 2018: Phys. Rev. Lett. 121, 180502) and will generate a gate with K loops by choosing:

$\begin{matrix} {\Delta = {4\Omega_{g}K^{1/2}{\left\{ {\frac{\left\lbrack {J_{4}\left( \frac{4\Omega_{\mu}}{\delta} \right)} \right\rbrack^{2}}{j} + \frac{\left\lbrack {J_{8}\left( \frac{4\Omega_{\mu}}{\delta} \right)} \right\rbrack^{2}}{j + 1}} \right\}^{1/2}.}}} & (34) \end{matrix}$

Since J₄(4Ω_(μ)/δ) and J₈(4Ω_(μ)/δ) are independent functions, the relative amplitudes of their effective tones can be optimized by setting the value of Ω_(μ)/δ. For example, when j=1, the gate can be robust to gate duration errors when J₈(4Ω_(μ)/δ)/J₄(4Ω_(μ)/δ)=−1 or when J₈(4Ω_(μ)/δ)/J₄(4Ω_(μ)/δ)=−2. In certain implementations, these gates still exhibit sensitivity to time-varying qubit frequency shifts.

In certain implementations, the gate utilizes intrinsic dynamical decoupling (IDD) to reduce (e.g., eliminate) the sensitivity to time-varying qubit frequency shifts. As described herein, when the value of 4Ω_(μ)/δ is set to one of the zeros of the J₀ Bessel function, the expression in Eq. (17) goes to zero and the qubit frequency shifts do not contribute to the dynamics. The off-resonant terms dropped from Eq. (17) all oscillate at integer multiples of δ (e.g., where ε, ω_(ε)<<δ), their effect averages to zero. In certain implementations, δ, ω_(g), and ω_(r) are selected such that (i) Eqs. (32) are met for a particular value of j, and (ii) 4Ω_(μ)/δ≃8.65, such that the gate is operated at the third zero of J₀(4Ω_(μ)/δ) (e.g., the third IDD point) at which J₈(4Ω_(μ)/δ)/J₄(4Ω_(μ)/δ)≃−1.22 (see FIG. 5C), which can be referred to as the IDD-j gate.

FIG. 5B shows the phase space trajectory of an IDD-2 gate in accordance with certain implementations described herein in comparison to that of a single-tone gate corresponding to the J₂ resonance at the first IDD point (which can be referred to as IDD-single). While the phase space trajectories of IDD-single and IDD-1 gates are not completely centered on the origin, those of IDD-j gates for j≥2 in accordance with certain implementations described herein are centered on the origin, resulting in less time-averaged spin-motion entanglement, and consequently less impact of motional decoherence on gate fidelity. In the following numerical calculations with regard to IDD: Ω_(g)/2π=1 kHz and ω_(r)/2π=6.5 MHz. For the IDD-single gate, ω_(g)/2π=5 MHz and for the IDD-j gates, ω_(g)˜ω_(r)/3.

In addition to the IDD effect, the IDD-j gates of certain implementations can be made insensitive to static qubit frequency shifts. With an effective interaction of the form σ_(z,1)σ_(z,2), the qubit frequency shifts commute with the ZZ gate operation. As a result, the effect of static qubit frequency shifts can be removed (when ε<<δ) by performing a K=2 loop gate with an instantaneously applied qubit π rotation in between loops (e.g., a Walsh modulation of index 1; see, e.g., Hayes 2012).

FIG. 6A shows numerical simulations (including all terms in Eq. (26)) of the impact of static shifts ε on the gate infidelity 1−

, which is below the 10⁻⁴ level for ε≲300Ω_(g), in accordance with certain implementations described herein. The dashed lines correspond to the IDD-single gate and the solid lines correspond to the IDD-2 gate with a gradient strength Ω_(g). The gate fidelity

can be defined as

≡

Φ⁺|{circumflex over (ρ)}(t_(G))|Φ⁺

, where |Φ⁺

≡1/√{square root over (2)}(|↓↓

)+i|↑↑

is the target qubit state, when starting in |↓↓

and performing global π/2 rotations perpendicular to z immediately before and after the gate. This technique breaks down as ε approaches δ, such that the off-resonant terms become non-negligible.

FIG. 6B shows numerical simulations of the sensitivity of the gate infidelity 1−

to oscillating qubit frequency shifts (plotted as a function ω_(ε), assuming a shift amplitude ε=Ω_(g)/5) in accordance with certain implementations described herein. For 1−

<<1, the infidelity scales as ε². The numerical simulations shown in FIGS. 6A and 6B use a 20 μs Blackman envelope to shape the rising and falling edges of the gradient pulses and the microwave pulses. For example, the microwave pair can be turned on first, and the gradient can be ramped up after the microwaves reach steady state, and ramped down in the reverse order at the end of the pulses. The gate speed is linear in the gradient strength, while it depends on the microwave amplitude as the argument of the two Bessel functions J₄ and J₈ (see Eq. (33)), which can cause complicated undesired dynamics during the microwave ramp if the gradient is already present. The errors seen in FIG. 6B occur during the microwave rise and fall times, when Ω_(μ) is not at the IDD point, such that the qubits are vulnerable to frequency fluctuations of the form in Ĥ_(z)(t). Qubit shifts that oscillate at or near nδ (for integer n) appear in the bichromatic interaction picture as static error terms ∝Ŝ_(z) (for even n) or ∝Ŝ_(y) (for odd n). Experimentally, qubit frequency fluctuations near nδ can arise from residual magnetic fields at ω_(g) from the currents generating the gradient. Choosing the J₄ and J₈ resonances to implement the gate makes n even, and so the resulting errors are ∝Ŝ_(z) and can be removed as described herein.

In certain implementations, gates with multiple blue and red sideband pairs have reduced sensitivity to motional frequency offsets. For example, an IDD-j gate is a linear superposition of a K=j and a K=j+1 loop gate, each with amplitudes of opposite signs. A motional frequency offset ν shifts the secular frequency such that ω_(r)→ω_(r)+ν, resulting in a residual displacement in phase space at the end of the gate. With nonzero ν, the superposed gates experience opposite displacements in phase space which coherently cancel each other. FIG. 7A is a logarithmic plot of the gate infidelity 1−

as a function of the motional frequency offset ν for both of the qubit-frequency shift-resistant gates of FIGS. 6A and 6B, with a fixed gradient Ω_(g), in accordance with certain implementations described herein. As shown in FIG. 7A, the IDD-2 gate has a lower sensitivity to ν than does the IDD-single gate.

FIG. 7B plots the Bell state infidelity for various IDD gates in accordance with certain implementations described herein. FIG. 7B shows the IDD-single gate performed with Walsh modulation of index 3 or 4 (K=8 and K=16, respectively) and the IDD-2 gate and the IDD-3 gate with K=1 or K=6, plotted against a dimensionless motional offset ν/Ω_(g) normalized by a dimensionless gate duration t_(G)Ω_(g)/2π. With this normalization, the different gates fall on approximately the same curve of sensitivity to motional frequency offsets, as do IDD-single gates for small values of (ν/Ω_(g))/(t_(G)Ω_(g)/2π). As the gate-time-normalized motional frequency offset becomes larger, the IDD-j gates have higher fidelity than single tone gates following Walsh sequences, as shown in FIG. 7B. FIG. 7B shows that the coherent error cancellation can provide reduced sensitivity to ν by increasing either the number of loops K or the order j of the gate. The infidelity 1−

due to an offset ν can remain constant for an increased ν if t_(G) is also increased proportionally, whether the increased t_(G) is associated with more loops K or larger values of j.

FIG. 8A is a plot of the gate infidelity due to motional heating versus normalized gate speed 2πΩ_(g)t_(G) in accordance with certain implementations described herein. Point labels indicate the number of phase space loops K. For a given gradient strength Ω_(g) and heating rate {dot over (n)}=Ω_(g)/100π, the error of agate scales with 1/t_(G). For a given t_(G), the IDD-2 gate (solid line) and the IDD-1 gate (light dashed line) outperform the IDD-single gate (dark dashed line). Higher-order IDD-j gates for j≥3 are similar to the IDD-2 gate. FIG. 8B is a plot of the same calculations, but for motional dephasing instead of motional heating, assuming a motional dephasing rate of Γ_(d)=Ω_(g)/100π, in accordance with certain implementations described herein. These decoherence mechanisms can be treated as Markovian, using a Lindblad formalism. For these calculations, Eq. (33) was used to calculate the infidelity, which gives the same motional decoherence effect as the full integration of Eq. (26).

FIGS. 8A and 8B show the increased robustness to heating and motional dephasing of the IDD-j gates in accordance with certain implementations described herein. In certain implementations, geometric phase gates can be made less sensitive to motional heating by performing more phase space loops, with 1−

scaling ∝1/t_(G). Both calculations of FIGS. 8A and 8B show that, while better than the IDD-single gate, the IDD-1 gate is not as robust as is the IDD-2 gate, which can be understood because the IDD-1 trajectory is not centered on the origin of phase space. For j>2, however, there is not a significant improvement of the fidelity versus t_(G) relative to j=2, because the phase space trajectory of the IDD-2 gate is already centered on the origin, thus saturating improvement to the time-averaged spin-motion entanglement.

Example Surface Electrode Trapped-Ion Quantum Logic Gate

FIG. 9 schematically illustrates an example surface electrode configuration (developed and disclosed by Ospelkaus 2008) for an example trapped-ion quantum logic gate 100 compatible with certain implementations described herein. In certain implementations, the gate 100 of FIG. 9 is configured for operation as a {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate (e.g., a {circumflex over (σ)}_(y)⊗{circumflex over (σ)}_(y) gate or “YY gate”), while in certain other implementations, the gate 100 of FIG. 9 is configured for operation as provides a {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate (e.g., a “ZZ gate”).

The example gate 100 of FIG. 9 comprises a plurality of electrodes 110 at a surface (e.g., in the yz plane) of a substrate (not shown) having a surface normal generally along the x-direction. Each of the electrodes 110 extends generally along the y-direction, as shown in FIG. 9 . The plurality of electrodes 110 comprises a first electrode 110 a, two second electrodes 110 b, a third electrode 110 c, a fourth electrode 110 d, and a plurality of segmented fifth electrodes 110 e. The first electrode 110 a is between the two second electrodes, and the first and second electrodes 110 a, 110 b are between the third electrode 110 c and the fourth electrode 110 d. The first, second, third, and fourth electrodes 110 a, 110 b, 110 c, 110 d are between a first set of the fifth electrodes 110 e and a second set of the fifth electrodes 110 e.

In certain implementations, a first potential that is static (e.g., constant) is applied to the first electrode 110 a, a second potential that oscillates at a radio frequency (e.g., rf potential) is applied to each of the two second electrodes 110 b, and third, fourth, and fifth potentials that are static (e.g., constant) are applied to the third, fourth, and fifth electrodes 110 c, 110 d, 110 e, respectively, such that the third, fourth, and fifth electrodes 110 c, 110 d, 110 e provide confinement along the y-direction. The oscillating currents in the first, third, and fourth electrodes 110 a, 110 c, 110 d can be used to implement single-qubit rotations and entangling gates. In certain implementations, the electrodes 110 have a uniform distribution of current in the surface, and induced currents in neighboring trap electrodes 110, which can alter the magnetic fields and their gradients at the position of the ion 120, are reduced (e.g., minimized).

In certain implementations, at least one ion 120 is trapped (e.g., at an rf pseudopotential null line) at a distance do above the trap plane (e.g., the yz plane). For example, the at least one ion 120 can comprise one ion 120 (e.g., ⁹Be⁺) having mass m or a plurality of N ions 120 (e.g., two, three, four, or more ⁹Be⁺ ions), each ion 120 having mass m, and the plurality of N ions 120 generally aligned with one another along the y direction. Examples of other ions that can be used include, but are not limited to, Mg⁺, Ca⁺, Sr⁺, Ba⁺, Zn⁺, Cd⁺, Hg⁺, and Yb⁺. Each ion 120 can comprise two internal states with |↑

and |↓

that compose a qubit with a transition frequency ω₀. The z direction can be the quantization axis, provided by a static magnetic field B₀{right arrow over (e)}_(z), with the n^(th) ion 120 having a displacement q_(n) (along the x or z directions) relative to its equilibrium position (q_(n)=0).

In certain implementations, at least some of the plurality of electrodes 110 can have widths along the z-direction that are configured such that antiparallel currents I(t)=Ĩ cos(ωt+φ) through the third and fourth electrodes 110 c, 110 d provide a magnetic field (e.g., B _(x)=1.5×10⁻⁷ I(t)/d₀ T at the ion 120 to implement rotations, where I and d₀ are expressed in amperes and meters, respectively). For example, the first, third, and fourth electrodes 110 a, 110 c, 110 d have a width along the z-direction that is (5/4)d₀ and the second electrodes can have a width along the z-direction that is (39/40)d₀.

In certain implementations, for d₀=30 μm and using a field-independent ⁹Be⁺ ion 120 as the qubit, (B₀=12 mT with |↑

=|F=1, m_(F)=1

and |↓

=|F=2, m_(F)=0

, μ_(x↑↓)=0.48μ_(B), where μ_(B) is the Bohr magneton), a carrier π pulse can be generated in 1 μs with Ĩ=15 mA. For multiqubit gates, an oscillating current I(t) can be applied through the first electrode 110 a to produce a field B_(z)(t) at the ion 120. Two currents at −2.5 I(t) can be applied through the third and fourth electrodes 110 c, 110 d to produce a magnetic field −B_(z)(t) at the ion 120, thereby nulling the magnetic field while all three currents provide field gradients ∂B_(x)(t)/∂z=∂B_(z)(t)/∂x=2.5×10⁻⁷ I(t)/d₀ ² T/m for motional-state excitation and multi-qubit gates. For example, {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate can be implemented with the first-order field-insensitive ⁹Be⁺ ion 120 as the qubit and d₀=30 μm, with a radial z “rocking” or center-of-mass (COM) mode with ω_(j)=2π×5 MHz in τ=20 μs with Ĩ=1.7 A [τ∝(mω_(j))^(1/2)/(∂B_(x)/∂z)]. For another example, a {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate can be implemented with the first-order field-insensitive ⁹Be⁺ ion 120 as the qubit and the states |↑

=|F=2, m_(F)=2

and |↓

=|F=2, m_(F)=0γ (μ_(z↑↑)=−1.0μ_(B); μ_(z↓↓)=0.26μ_(B)), reached from the field-independent qubit manifold with one microwave pulse. For a radial x rocking or COM mode, the {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate can be realized in the same time with Ĩ=1.3 A.

In certain implementations, an entangling gate comprises at least one ion 120, a magnetic field gradient, and two microwave fields. The internal states of the at least one ion 120 are coupled via a motional mode having a frequency ω_(r) and all other motional modes are sufficiently detuned from ω_(r) such that they do not coupled to the spins of the at least one ion 120. The magnetic field gradient is a gradient, along the motional mode, of a component (e.g., along the x, y, or z direction) of a magnetic field, and has a time dependence f(t). For example, f(t) can be constant or can be sinusoidally oscillating. Each of the two microwave fields has a respective frequency that is detuned from the qubit transition frequency ω₀ of the at least one ion 120 by a frequency difference δ (e.g., ω₁=ω₀+δ and ω₂=ω₀−δ), with δ<<ω₀ (e.g., δ/ω₀ is less than 0.05; δ/ω₀ is less than 0.01). The at least one ion 120 has a Rabi frequency Ω_(μ) corresponding to fluctuations in the populations of the two states of the at least one ion 120 due to the microwave fields. The at least one ion 120 also has a Rabi frequency Ω_(g) corresponding to fluctuations in the populations of the two states of the at least one ion 120 due to the magnetic field gradient.

In certain implementations, the values of Ω_(g) and δ and are selected such that the quantity Ω_(g)/δ is small (e.g., in a range between zero and 5×10⁻²; in a range between zero and 1×10⁻²; in a range between zero and 1×10⁻³). For example, Ω_(g) can be on the order of kHz (e.g., Ω_(g)/2π is in a range of 0.1 kHz to 10 kHz) and δ can be on the order of MHz or GHz (e.g., δ/2π is in a range of 0.1 MHz to 10 GHz). In certain implementations, the values of Ω_(μ) and δ and are selected such that the quantity 4Ω_(μ)/δ is close to a zero of the J₀ Bessel function. For example, the quantity 4Ω_(μ)/δ can in a range within 10% of the first zero of the J₀ Bessel function (2.405), a range within 10% of the second zero of the J₀ Bessel function (5.520), or a range within 10% of the third zero of the J₀ Bessel function (8.654). In certain implementations, the values of Ω_(μ) and δ and are selected such that the quantity 4Ω_(μ)/δ is close to (e.g., within 10%) of the value at which an absolute value of the J_(n) Bessel function, where n≥1, is at a maximum. In certain implementations, the values of Ω_(μ) and δ are selected such that J₈(4Ω_(μ)/δ)/J₄(4Ω_(μ)/δ) is close to (e.g., within 10%) of −1, −1.22, or −2.

Conditional language, such as “can,” “could,” “might,” or “may,” unless specifically stated otherwise, or otherwise understood within the context as used, is generally intended to convey that certain implementations include, while other implementations do not include, certain features, elements, and/or steps. Thus, such conditional language is not generally intended to imply that features, elements, and/or steps are in any way required for one or more implementations.

Conjunctive language such as the phrase “at least one of X, Y, and Z,” unless specifically stated otherwise, is to be understood within the context used in general to convey that an item, term, etc. may be either X, Y, or Z. Thus, such conjunctive language is not generally intended to imply that certain implementations require the presence of at least one of X, at least one of Y, and at least one of Z.

Language of degree, as used herein, such as the terms “approximately,” “about,” “generally,” and “substantially,” represent a value, amount, or characteristic close to the stated value, amount, or characteristic that still performs a desired function or achieves a desired result. For example, the terms “approximately,” “about,” “generally,” and “substantially” may refer to an amount that is within ±10% of, within ±5% of, within ±2% of, within ±1% of, or within ±0.1% of the stated amount. As another example, the terms “generally parallel” and “substantially parallel” refer to a value, amount, or characteristic that departs from exactly parallel by ±10 degrees, by ±5 degrees, by ±2 degrees, by ±1 degree, or by ±0.1 degree, and the terms “generally perpendicular” and “substantially perpendicular” refer to a value, amount, or characteristic that departs from exactly perpendicular by ±10 degrees, by ±5 degrees, by ±2 degrees, by ±1 degree, or by ±0.1 degree.

Various configurations have been described above. Although this invention has been described with reference to these specific configurations, the descriptions are intended to be illustrative of the invention and are not intended to be limiting. Various modifications and applications may occur to those skilled in the art without departing from the true spirit and scope of the invention. Thus, for example, in any method or process disclosed herein, the acts or operations making up the method/process may be performed in any suitable sequence and are not necessarily limited to any particular disclosed sequence. Features or elements from various implementations and examples discussed above may be combined with one another to produce alternative configurations compatible with implementations disclosed herein. Various aspects and advantages of the implementations have been described where appropriate. It is to be understood that not necessarily all such aspects or advantages may be achieved in accordance with any particular implementation. Thus, for example, it should be recognized that the various implementations may be carried out in a manner that achieves or optimizes one advantage or group of advantages as taught herein without necessarily achieving other aspects or advantages as may be taught or suggested herein. 

What is claimed is:
 1. A method of operating a trapped-ion quantum logic gate, the method comprising: providing a trapped-ion quantum logic gate, the gate comprising: at least one ion having two internal states and forming a qubit having a qubit transition frequency ω₀; a magnetic field gradient; and two microwave fields, each having a respective frequency that is detuned from the qubit transition frequency ω₀ by frequency difference δ, the at least one ion having a Rabi frequency Ω_(μ) due to the two microwave fields and a Rabi frequency Ω_(g) due to the magnetic field gradient; and applying the magnetic field gradient and the two microwave fields to the at least one ion such that a quantity Ω_(g)/δ is in a range between zero and 5×10⁻².
 2. The method of claim 1, wherein the two internal states of the at least one ion are coupled via a motional mode having a frequency Ω_(r), and the magnetic field gradient comprises a gradient, along the motional mode, of a component of a magnetic field, and has a time dependence f(t).
 3. The method of claim 2, wherein the time dependence f(t) is constant.
 4. The method of claim 2, wherein the time dependence f(t) comprises a sinusoidal oscillation.
 5. The method of claim 1, wherein δ/ω₀ is less than 0.05.
 6. The method of claim 1, wherein δ/ω₀ is less than 0.01.
 7. The method of claim 1, wherein Ω_(g)/2π is in a range of 0.1 kHz to 10 kHz and δ/2π is in a range of 0.1 MHz to 10 GHz.
 8. The method of claim 7, further comprising applying the magnetic field gradient and the two microwave fields to the at least one ion such the at least one ion experiences intrinsic dynamical decoupling.
 9. The method of claim 8, wherein a quantity 4Ω_(μ)/δ is within 10% of a zero of a J₀ Bessel function.
 10. The method of claim 8, wherein the quantity 4Ω_(μ)/δ is within 10% of the value at which an absolute value of the J_(n) Bessel function, where n≥1, is at a maximum.
 11. The method of claim 8, wherein the values of Ω_(μ) and δ are selected such that J₈(4Ω_(μ)/δ)/J₄(4Ω_(μ)/δ) is within 10% of −1, −1.22, or −2.
 12. A trapped-ion quantum logic gate comprising: at least one ion having two internal states and forming a qubit having a qubit transition frequency ω₀; a magnetic field gradient; and two microwave fields, each having a respective frequency that is detuned from the qubit transition frequency ω₀ by frequency difference δ, the at least one ion having a Rabi frequency Ω_(μ) due to the two microwave fields and a Rabi frequency Ω_(g) due to the magnetic field gradient, the magnetic field gradient and the two microwave fields configured such that a quantity Ω_(g)/δ is in a range between zero and 5×10⁻².
 13. The trapped-ion quantum logic gate of claim 12, wherein the gate comprises a {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate.
 14. The trapped-ion quantum logic gate of claim 12, wherein the gate comprises a {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate.
 15. The trapped-ion quantum logic gate of claim 12, further comprising a surface electrode trap in which the at least one ion is confined, wherein the two internal states of the at least one ion are coupled via a motional mode having a frequency ω_(r), and the magnetic field gradient comprises a gradient, along the motional mode, of a component of a magnetic field, and has a time dependence f(t)=cos(ω_(g)t), the at least one ion having resonances when δ is within 10% of an integer multiple of |ω_(r)±ω_(g)|.
 16. The trapped-ion quantum logic gate of claim 15, wherein δ is within 10% of (ω_(r)−ω_(g)), and the gate comprises a {circumflex over (σ)}_(ϕ)⊗{circumflex over (σ)}_(ϕ) gate with a Rabi frequency equal to Ω_(g)J₁(4Ω_(μ)/δ), where J₁ is the first Bessel function.
 17. The trapped-ion quantum logic gate of claim 15, wherein δ is within 10% of |ω_(r)−ω_(g)|/2, and the gate comprises a {circumflex over (σ)}_(z)⊗{circumflex over (σ)}_(z) gate with a Rabi frequency equal to Ω_(g)J₂(4Ω_(μ)/δ), where J₂ is the second Bessel function.
 18. The trapped-ion quantum logic gate of claim 15, wherein the values of Ω_(μ) and δ and are selected such that the quantity 4Ω_(μ)/δ is in a range within 10% of the first zero of the J₀ Bessel function (2.405), a range within 10% of the second zero of the J₀ Bessel function (5.520), or a range within 10% of the third zero of the J₀ Bessel function (8.654).
 19. The trapped-ion quantum logic gate of claim 12, wherein the values of Ω_(μ) and δ are selected such that J₈(4Ω_(μ)/δ)/J₄(4Ω_(μ)/δ) is within 10% of −1, −1.22, or −2. 